RGEs in EFTs

Last updated Aug 5, 2023

• Rework

Consider a general EFT

$$\mathcal{L} = \sum_i c_i(\mu,\Lambda) \frac{\mathcal{O}_i}{\Lambda^{d_i - 4}}$$

where we only have a single operator per dimension (so no operator mixing)

A product of Wilson coefficients and Operators must always be independent of the scale $\mu$ (since it is a physical observable) $$\mu \frac{\text{d}}{\text{d}\mu} c_i(\mu,\Lambda)<\mathcal{O}(\mu)>_R=0.$$ $<\cdots>_R$ denotes the renormalized greens function.

The bare Greens function $<\mathcal{O}_i> = Z_i(\epsilon,\mu)<\mathcal{O}_i(\mu)>_R$ is also independent of $\mu$ so that from $\frac{\text{d}}{\text{d}\mu}<\mathcal{O}_i>=0$ we get the RG equation

$$(\mu \frac{\text{d}}{\text{d}\mu} + \gamma_{O_i})<\mathcal{O}_i(\mu)>R = 0,$$ where $\gamma{O_i} = \frac{\mu}{Z_i}\frac{\text{d}Z_i}{\text{d}\mu}$ is the anomalous dimension of the operator $\mathcal{O}_i$.

Using this result we can derive the RG equation for the Wilson coefficients $c_i$

$$\mu\frac{\text{d}}{\text{d}\mu}c_i(\mu,\Lambda) = \gamma_{O_i}(\alpha_R)c_i(\mu,\Lambda)$$

The RG equation can be solved to obtain the evolution of $c_i(\mu)$ from $\mu_0$ to $\mu$

$$c_i(\mu) = c_i(\mu_0) e^{\int^{\alpha_R(\mu)}{\alpha_R(\mu_0)} \frac{\text{d}\alpha}{\alpha}\frac{\gamma{O_i}}{\beta(\alpha)}}$$

Here $\gamma_{O_i}$ and the beta function $\beta$ can be expanded in terms of the coupling constant $\alpha$

If we choose $\mu_0 = m$ to evolve a Wilson coefficient down from a heavy mass threshold. We expand the Wilson coefficent in terms of the coupling constant $$c_i(m) = c^{(0)}_i + \frac{\alpha_R(m)}{4\pi}c^{(1)}_i.$$

• In Leading logarithmic accuracy (LL) we have $$c_i(\mu) = c^{(0)}i\left(\frac{\alpha_R(\mu)}{\alpha_R(m)}\right)^{-\frac{\gamma^{(0)}}{2\beta_0}}$$ LL accuracy resums terms of the type $1 + \sum{n>0}…(\alpha \log\frac{\mu}{m})^n$

• In Next to leading logarithmic accuracy (NLL) we have $$c_i(\mu) = c^{(0)}_i\left(\frac{\alpha_R(\mu)}{\alpha_R(m)}\right)^{-\frac{\gamma^{(0)}}{2\beta_0}}\left[1 + \frac{\alpha_R(m)}{4\pi}\frac{c^{(1)}_i}{c^{(0)}i} - (\frac{\gamma^{(1)}}{2\beta_0} - \frac{\beta_i}{2\beta^2_0}\gamma^{(0)})\frac{\alpha_R(\mu)-\alpha_R(m)}{4\pi} + \mathcal{O}(\alpha^3_R)\right]$$ NLL accuracy resums terms of the type $\alpha + \sum{n>0}…\alpha (\alpha \log\frac{\mu}{m})^n$

The anomalous dimension can be computed by $$\gamma_{O_i}(\alpha) = -2\alpha\frac{\partial}{\partial\alpha}Z^{(i)}i$$ where $$Z_i(\epsilon,\mu) = 1 + \sum^\infty{n=1}\frac{Z^{(n)}_i}{\epsilon^n}$$ in the MS scheme

Renormalization can lead to operator mixing $$<\mathcal{O}i> = \sum_i Z{ij}(\epsilon,\mu)<\mathcal{O}_j(\mu)>R$$ where $Z{ij}$ is the renormalization matrix. The anomalous dimension matrix $\gamma_O$ follows as $$\gamma_O = Z^{-1}\mu\frac{d}{d\mu}Z$$

We get the RG equations $$(\mu\frac{d}{d\mu} + \gamma_O)<\mathcal{O}>_R=0$$ $$(\mu\frac{d}{d\mu} + \gamma^T_O)c=0$$ where $<\mathcal{O}>$ and $c$ are vectors of the operators and coefficients respectively.

To solve the RG equation we can diagonalize $\gamma_0$, which also transforms the coefficents $$U^{-1}\gamma^T_OU = \tilde{\gamma_O}, U^{-1}c = \tilde{c}.$$ The solution is then $$\tilde{c}j(\mu) = \tilde{c}j(\mu_0)\exp\left(\int^{\alpha(\mu)}{\alpha(\mu_0)}\frac{d\alpha}{\alpha}\frac{\tilde{\gamma{O_j}}(\alpha)}{\beta(\alpha)}\right),$$ where $\tilde{\gamma_{O_j}}(\alpha)$ is the $j$th diagonal element. Afterwards we can transfrom the coefficients back into the non-diagonalized basis.

1. $c_i(\Lambda_H,\Lambda_H)$ is calculated to some order by matching Green’s functions at the scale $\Lambda_H$
2. The anomalous dimension of $c_i$ is computed at the corresponding order(1 loop for $c_i$ at LO gives (LL), 2 loops for $c_i$ at NLO gives NLL)
3. Solve the RG equation for $c_i$ and run down to $\Lambda_L$ to get $c_i(\Lambda_L,\Lambda_H)$ and $\mathcal{O}_i(\Lambda_L,\Lambda_L)$.